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Theorem List for Intuitionistic Logic Explorer - 3401-3500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsnidb 3401 A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)

Theoremsnid 3402 A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 31-Dec-1993.)

Theoremvsnid 3403 A setvar variable is a member of its singleton (common case). (Contributed by David A. Wheeler, 8-Dec-2018.)

Theoremelsn2g 3404 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that , rather than , be a set. (Contributed by NM, 28-Oct-2003.)

Theoremelsn2 3405 There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that , rather than , be a set. (Contributed by NM, 12-Jun-1994.)

Theoremmosn 3406* A singleton has at most one element. This works whether is a proper class or not, and in that sense can be seen as encompassing both snmg 3486 and snprc 3435. (Contributed by Jim Kingdon, 30-Aug-2018.)

Theoremralsnsg 3407* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremralsns 3408* Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremrexsns 3409* Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.)

TheoremrexsnsOLD 3410* Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) Obsolete as of 22-Aug-2018. Use rexsns 3409 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Theoremralsng 3411* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrexsng 3412* Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)

Theoremexsnrex 3413 There is a set being the element of a singleton if and only if there is an element of the singleton. (Contributed by Alexander van der Vekens, 1-Jan-2018.)

Theoremralsn 3414* Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)

Theoremrexsn 3415* Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)

Theoremeltpg 3416 Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)

Theoremeltpi 3417 A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)

Theoremeltp 3418 A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Apr-1994.) (Revised by Mario Carneiro, 11-Feb-2015.)

Theoremdftp2 3419* Alternate definition of unordered triple of classes. Special case of Definition 5.3 of [TakeutiZaring] p. 16. (Contributed by NM, 8-Apr-1994.)

Theoremnfpr 3420 Bound-variable hypothesis builder for unordered pairs. (Contributed by NM, 14-Nov-1995.)

Theoremralprg 3421* Convert a quantification over a pair to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrexprg 3422* Convert a quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremraltpg 3423* Convert a quantification over a triple to a conjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrextpg 3424* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremralpr 3425* Convert a quantification over a pair to a conjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrexpr 3426* Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremraltp 3427* Convert a quantification over a triple to a conjunction. (Contributed by NM, 13-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrextp 3428* Convert a quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015.)

Theoremsbcsng 3429* Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremnfsn 3430 Bound-variable hypothesis builder for singletons. (Contributed by NM, 14-Nov-1995.)

Theoremcsbsng 3431 Distribute proper substitution through the singleton of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Theoremdisjsn 3432 Intersection with the singleton of a non-member is disjoint. (Contributed by NM, 22-May-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Wolf Lammen, 30-Sep-2014.)

Theoremdisjsn2 3433 Intersection of distinct singletons is disjoint. (Contributed by NM, 25-May-1998.)

Theoremdisjpr2 3434 The intersection of distinct unordered pairs is disjoint. (Contributed by Alexander van der Vekens, 11-Nov-2017.)

Theoremsnprc 3435 The singleton of a proper class (one that doesn't exist) is the empty set. Theorem 7.2 of [Quine] p. 48. (Contributed by NM, 5-Aug-1993.)

Theoremr19.12sn 3436* Special case of r19.12 2422 where its converse holds. (Contributed by NM, 19-May-2008.) (Revised by Mario Carneiro, 23-Apr-2015.)

Theoremrabsn 3437* Condition where a restricted class abstraction is a singleton. (Contributed by NM, 28-May-2006.)

Theoremrabrsndc 3438* A class abstraction over a decidable proposition restricted to a singleton is either the empty set or the singleton itself. (Contributed by Jim Kingdon, 8-Aug-2018.)
DECID

Theoremeuabsn2 3439* Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016.)

Theoremeuabsn 3440 Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.)

Theoremreusn 3441* A way to express restricted existential uniqueness of a wff: its restricted class abstraction is a singleton. (Contributed by NM, 30-May-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

Theoremabsneu 3442 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.)

Theoremrabsneu 3443 Restricted existential uniqueness determined by a singleton. (Contributed by NM, 29-May-2006.) (Revised by Mario Carneiro, 23-Dec-2016.)

Theoremeusn 3444* Two ways to express " is a singleton." (Contributed by NM, 30-Oct-2010.)

Theoremrabsnt 3445* Truth implied by equality of a restricted class abstraction and a singleton. (Contributed by NM, 29-May-2006.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

Theoremprcom 3446 Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.)

Theorempreq1 3447 Equality theorem for unordered pairs. (Contributed by NM, 29-Mar-1998.)

Theorempreq2 3448 Equality theorem for unordered pairs. (Contributed by NM, 5-Aug-1993.)

Theorempreq12 3449 Equality theorem for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq1i 3450 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq2i 3451 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq12i 3452 Equality inference for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq1d 3453 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq2d 3454 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theorempreq12d 3455 Equality deduction for unordered pairs. (Contributed by NM, 19-Oct-2012.)

Theoremtpeq1 3456 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Theoremtpeq2 3457 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Theoremtpeq3 3458 Equality theorem for unordered triples. (Contributed by NM, 13-Sep-2011.)

Theoremtpeq1d 3459 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtpeq2d 3460 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtpeq3d 3461 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtpeq123d 3462 Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.)

Theoremtprot 3463 Rotation of the elements of an unordered triple. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremtpcoma 3464 Swap 1st and 2nd members of an undordered triple. (Contributed by NM, 22-May-2015.)

Theoremtpcomb 3465 Swap 2nd and 3rd members of an undordered triple. (Contributed by NM, 22-May-2015.)

Theoremtpass 3466 Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremqdass 3467 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremqdassr 3468 Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)

Theoremtpidm12 3469 Unordered triple is just an overlong way to write . (Contributed by David A. Wheeler, 10-May-2015.)

Theoremtpidm13 3470 Unordered triple is just an overlong way to write . (Contributed by David A. Wheeler, 10-May-2015.)

Theoremtpidm23 3471 Unordered triple is just an overlong way to write . (Contributed by David A. Wheeler, 10-May-2015.)

Theoremtpidm 3472 Unordered triple is just an overlong way to write . (Contributed by David A. Wheeler, 10-May-2015.)

Theoremtppreq3 3473 An unordered triple is an unordered pair if one of its elements is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)

Theoremprid1g 3474 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)

Theoremprid2g 3475 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by Stefan Allan, 8-Nov-2008.)

Theoremprid1 3476 An unordered pair contains its first member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)

Theoremprid2 3477 An unordered pair contains its second member. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)

Theoremprprc1 3478 A proper class vanishes in an unordered pair. (Contributed by NM, 5-Aug-1993.)

Theoremprprc2 3479 A proper class vanishes in an unordered pair. (Contributed by NM, 22-Mar-2006.)

Theoremprprc 3480 An unordered pair containing two proper classes is the empty set. (Contributed by NM, 22-Mar-2006.)

Theoremtpid1 3481 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremtpid2 3482 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremtpid3g 3483 Closed theorem form of tpid3 3484. (Contributed by Alan Sare, 24-Oct-2011.)

Theoremtpid3 3484 One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Theoremsnnzg 3485 The singleton of a set is not empty. (Contributed by NM, 14-Dec-2008.)

Theoremsnmg 3486* The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremsnnz 3487 The singleton of a set is not empty. (Contributed by NM, 10-Apr-1994.)

Theoremsnm 3488* The singleton of a set is inhabited. (Contributed by Jim Kingdon, 11-Aug-2018.)

Theoremprmg 3489* A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)

Theoremprnz 3490 A pair containing a set is not empty. (Contributed by NM, 9-Apr-1994.)

Theoremprm 3491* A pair containing a set is inhabited. (Contributed by Jim Kingdon, 21-Sep-2018.)

Theoremprnzg 3492 A pair containing a set is not empty. (Contributed by FL, 19-Sep-2011.)

Theoremtpnz 3493 A triplet containing a set is not empty. (Contributed by NM, 10-Apr-1994.)

Theoremsnss 3494 The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)

Theoremeldifsn 3495 Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.)

Theoremeldifsni 3496 Membership in a set with an element removed. (Contributed by NM, 10-Mar-2015.)

Theoremneldifsn 3497 is not in . (Contributed by David Moews, 1-May-2017.)

Theoremneldifsnd 3498 is not in . Deduction form. (Contributed by David Moews, 1-May-2017.)

Theoremrexdifsn 3499 Restricted existential quantification over a set with an element removed. (Contributed by NM, 4-Feb-2015.)

Theoremsnssg 3500 The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.)

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